The two never competed.
While they both read Chakra patterns, Byakugan is good for Taijutsu users.
Sharingan has shown to be good with Genjutsu.
IMO, Byakugan will get its reawakening sometime to show that it doesn't lose by much to a Sharingan.
Who do you see winning between Neji and Sasuke? Sasuke! Why because his advantage is that he has MS. Neji is very adept at Taijutsu and we haven't seen whether there is an evolution for Byakugan(if you consider Hinata using the 1 eye Byakugan well...).